Distribution modulo one and ergodic theory
Jens Marklof
University of Bristol
http://www.maths.bristol.ac.uk
Prospects in Mathematics, Durham, December 2006
1
Measure rigidity is a branch of ergodic theory that has recently contributed to
the solution of some fundamental problems in number theory and mathematical
physics. Examples include
2
2
• quantitative versions of the Oppenheim conjecture m2
1 + m2 −
√
2 m2
3
(Eskin, Margulis & Mozes, Annals of Math 1998)
• spacings between the values of quadratic forms
√ 2
√ 2
(m1 − 2) + (m2 − 3) (Marklof, Annals of Math 2003)
√
2
m1 + 2 m2
2 (Eskin, Margulis & Mozes, Annals of Math 2005)
• quantum unique ergodicity for certain classes of hyperbolic surfaces
(Lindenstrauss, Annals of Math 2006)
• approach to Littlewood conjecture on the nonexistence of multiplicatively badly
approximable numbers
(Einsiedler, Katok & Lindenstrauss, Annals of Math 2006)
• Boltzmann-Grad limit of the periodic Lorentz gas
(Marklof & Strömbergsson, in preparation)
3
4
Ratner’s theorem is one of the central results in measure rigidity. It gives a
complete classification of measures invariant under unipotent flows. In this talk
I’ll discuss a few simple applications of Ratner’s theorem to the analysis of the
statistical properties of basic number-theoretic sequences:
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• fractional parts of nα, n = 1, 2, 3, . . .
(a classical, well understood sequence, usually studied by means of continued fractions; cf. Mazel-Sinai, Bleher, Greenman, Marklof)
√
• fractional parts of nα, n = 1, 2, 3, . . .
(as recently studied by Elkies and McMullen, Duke Math J 2004)
• fractional parts of n2α, n = 1, 2, 3, . . .
(Sinai, Pelegrinotti, Rudnick-Sarnak, Rudnick-Sarnak-Zaharescu, MarklofStrömbergsson,. . . )
6
For more details see
J. Marklof, Distribution modulo one and Ratner’s theorem
Proc. Montreal Summer School on Equidistribution in Number Theory 2005
(Springer, to appear)
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How to test randomness of point sequences mod 1
8
Consider an infinite triangular array of numbers on the circle T = R/Z ' [0, 1)
ξ11
ξ21 ξ22
...
...
...
ξN 1 ξN 2 . . . ξN N
...
...
...
Assume that each row is ordered, i.e., ξN j ≤ ξN (j+1); to simplify notation I’ll use
ξj instead of ξN j in the following.
Consider the number of elements in the interval [x0 − 2` , x0 + 2` ) mod 1,
SN (`) =
N
X
j=1
χ`(ξj ),
χ`(x) =
x − x0 + n
1,1)
[− 2
`
2
n∈Z
X
!
χ
1) ⊂
denotes the characteristic function of the interval [− 1
,
1
1
2 2
[− 2 , 2 )
Here χ
R.
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Uniform distribution mod 1. A sequence {ξj } is uniformly distributed mod 1 if
for every x0, `,
SN (`)
=`
N →∞ N
lim
Well known examples of u.d. sequences are nk α mod 1 (n = 1, 2, 3, . . .), for
any k ∈ N and α irrational (Weyl 1916)
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Poisson distribution. Let EN (k, L) be the probability of finding k elements in
L ) of size L/N (very small!)
the randomly shifted interval [x0, x0 + N
EN (k, L) := meas {x0 ∈ T : SN (`) = k}
We say the sequence {ξj } is Poisson distributed if
Lk −L
lim EN (k, L) =
e
N →∞
k!
This means {ξj } behaves like a generic realization of independent random variables mod 1.
Example. 2nα mod 1 is Poisson distributed for generic α (Rudnick & Zaharescu,
Forum Math 2002) and nk α mod 1 is conjectured to for k ≥ 2 by some people,
others disagree
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Distribution of gaps. A popular statistical measure is the distribution of gaps
sj = N (ξj+1 − ξj )
between consecutive elements. The gap distribution of {ξj } is
N
1 X
δ(s − sj )
PN (s) =
N j=1
and the question is whether it has a limiting distribution
PN (s) →w P (s),
i.e., for every bounded continuous function g : R → R,
lim
Z ∞
N →∞ 0
g(s)PN (s)ds =
Z ∞
0
g(s)P (s)ds.
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Note one can show
d2E(0, L)
PN (s) →w P (s) ⇐⇒ EN (0, L) → E(0, L) with
= P (L)
2
dL
In particular for Poisson distributed sequences P (s) = e−s.
13
Maple experiments
14
√
√
2
PN (s) for n 2 and n 2 vs. the exponential distribution
N = 6001
15
PN (s) for
√
n and
q √
n 2 vs. the exponential distribution
N = 10001
16
nα mod 1
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The number of elements in an interval of size ` = L/N and centered at x0 is
N X
X
!
N
(mα + n − x0)
1)
[− 1
,
L
2 2
m=1 n∈Z
!
X
m
χ(0,1]
=
χ[−L/2,L/2] N (mα + n − x0)
N
2
SN (`) =
=
(m,n)∈Z
X
χ
ψ (m, n − x0)
(m,n)∈Z2
1 α
0 1
!
N −1
0
0
N
!!
where χI denotes the characteristic function of the interval I ⊂ R and
ψ(x, y) = χ(0,1](x)χ[−L/2,L/2](y)
is the characteristic function of a rectangle.
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Let G = SL(2, R) n R2 with multiplication law
(M, ξ)(M 0, ξ 0) = (M M 0, ξM 0 + ξ 0),
where ξ, ξ 0 ∈ R2 are viewed as row vectors.
The function
F (M, ξ) =
X
ψ(mM + ξ)
m∈Z2
defines a function on G. Note that, with ψ as above, the sum is always finite, and
hence F is a piecewise constant function.
Note:
SN (`) = F (M, ξ)
for
M =
1 α
0 1
!
N −1
0
!
0
,
N
ξ = (0, −x0)M.
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Homogeneous spaces. The crucial observation is now that F is left-invariant
under the discrete subgroup Γ = SL(2, Z) n Z2
F (γg) = F (g)
∀
γ∈Γ
(this can be checked by an elementary calculation), and hence F may be viewed
as a piecewise constant function on the homogeneous space Γ\G.
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Dynamics on Γ\G. Right multiplication by
Φt =
e−t/2
0
0
et/2
!
!
,0
genrates a flow on Γ\G
Γg 7→ ΓgΦt.
Now observe that SN (`) is related to a function F on Γ\G evaluated along an
orbit of this flow:
SN (`) = F (g0Φt)
with t = 2 log N and initial condition
!
g0 = g0(α, x0) =
1 α
, (0, −x0)
0 1
!
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Equidistribution
Theorem. For any bounded, piecewise continuous f : Γ\G → R
b 1
1
1
lim
f (g0(α, y)Φt)dα dy =
f dµ.
t→∞ b − a a 0
µ(Γ\G) Γ\G
Z
Z
Z
This theorem can be proved by exploiting the mixing property of the flow Φt.
By chosing the right test function f , the following theorem (first proved by MazelSinai, and for P (s) by Bleher using different methods based on cont’d fractions)
is a direct corollary.
Values of nα in short intervals for random α
Theorem. For any L > 0,
meas{(α, x0) ∈ [a, b] × [0, 1] : SN (`) = k}
lim
= E(k, L),
N →∞
b−a
where
µ(g ∈ Γ\G : F (g) = k)
E(k, L) =
.
µ(Γ\G)
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Equidistribution II. Using Ratner’s theorem, one can in fact strengthen the previous results by showing:
Theorem. For any irrational x0 and any bounded, piecewise continuous f :
Γ\G → R
b
1
1
f (g0(α, x0)Φt)dα =
f dµ.
lim
t→∞ b − a a
µ(Γ\G) Γ\G
Z
Z
(Remarkably, A. Strömbergsson has recently proved this results by analytic techniques, without using Ratner’s theory.)
Hence we obtain the value distribution in intervals with fixed rather than random
x0 .
Theorem. For any L > 0 and x0 irrational,
meas{α ∈ [a, b] : SN (`) = k}
lim
= E(k, L),
N →∞
b−a
with the same E(k, L) as before.
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√
nα mod 1
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We are interested in the distribution of
N X
X
N √
SN (`) =
χ 11
( nα − x0 + m) .
[− 2 , 2 ) L
n=1 m∈Z
Similar to the previous section on can show that for α = 1
SN (`) ≈ F (g0(x)Φt)
with t = 2 log N and initial condition
!
g0(x) =
1 2x
, (x, x2)
0 1
!
Again
F (M, ξ) =
X
ψ(mM + ξ)
m∈Z2
but now
y
ψ(x, y) = χ(0,1](x)χ(−L,L]
x
which represents the characteristic function of a triangle.
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Equidistribution III
The following follows again from Ratner’s theorem (Elkies & McMullen 2004).
Theorem. For any bounded, piecewise continuous f : Γ\G → R
Z 1
1
lim
f (g0(x)Φt) dx =
f dµ.
t→∞ 0
µ(Γ\G) Γ\G
Z
This implies, as before:
Values of
√
n in short intervals
Theorem. For any L > 0,
lim meas{x ∈ [0, 1] : SN (`) = k} = E(k, L),
N →∞
where
µ(g ∈ Γ\G : F (g) = k)
E(k, L) =
.
µ(Γ\G)
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The above argument can be adapted for rational α but no longer works for irrational α. In fact heuristic arguments (see lecture notes) as well as numerical
√
experiments predict that then nα should be Poisson distributed, at least for
typical α.
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Ratner’s theorem
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Let
— G be a Lie group (e.g. SL(2, R) n R2)
— Γ be a discrete subgroup (e.g. SL(2, Z) n Z2)
— U a group generated by unipotent subgroups
Examples of such U we had discussed earlier are
(
!
!)
1 t
,0
0 1
t
(
!
1 0
, (0, t)
0 1
!)
t
(
!
1 2t
, (t, t2)
0 1
!)
t
Ratner’s theorem. Let ν be an ergodic, right-U -invariant probability measure on
Γ\G. Then there is a closed, connected subgroup H ⊂ G, and a point g ∈ Γ\G
such that
1. ν is H-invariant,
2. ν is supported on the orbit gH.
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Equidistribution IV
The equidistribution theorems we had used earlier are corollaries of the following
theorem, which in turn is a special case of a theorem by Shah (Proc. Indian Acad
Sci 1996).
Theorem. Suppose G contains a Lie subgroup H isomorphic to SL(2, R) (we
denote the corresponding embedding by ϕ : SL(2, R) → G), such that the set
Γ\ΓH is dense in Γ\G. Then, for any bounded continuous f : Γ\G → R
lim
Z b
t→∞ a
!
f ϕ
1 x
0 1
e−t/2
0
0
et/2
!!!
b−a
dx =
f dµ
µ(Γ\G) Γ\G
Z
where µ is the Haar measure of G.
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The general strategy of proof for statements of the above type is as follows.
1. Show that the sequences of probability measures νt defined by
νt(f ) =
Z b
1
f ϕ
b−a a
!
1 x
0 1
e−t/2
0
0
et/2
!!!
dx
is tight. Then, by the Helly-Prokhorov theorem, it is relatively compact, i.e., every
sequence of νt contains a convergent subsequence with weak limit ν, say.
3. Show that ν is invariant under a unipotent subgroup U ; in the present case,
(
U = ϕ
!!)
1 x
0 1
.
x∈R
4. Use a density argument to rule out measures concentrated on subvarieties
(exploit the assumption that Γ\ΓH is dense in Γ\G).
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